Relaxor ferroelectrics were discovered in the 1950s but many of their properties are not understood. In this review, we shall concentrate on materials such as PMN (PbMg_1/3Nb_2/3O₃), which crystallize in the cubic perovskite structure but with the Mg ion, charge 2+, and the Nb ion, charge 5+, randomly distributed over the B site of the perovskite structure. The peak of the dielectric susceptibility for relaxors is much broader in temperature than that of conventional ferroelectrics, while below the maximum of the susceptibility most relaxors remain cubic and show no electric polarization, unlike that observed for conventional ferroelectrics. Because of the large width of the susceptibility, relaxors are often used as capacitors. Recently, there have been many X-ray and neutron scattering studies of relaxors and the results have enabled a more detailed picture to be obtained. An important conclusion is that relaxors can exist in a random field state, as initially proposed by Westphal, Kleemann and Glinchuk, similar to that which has been studied for diluted antiferromagnets. If a relaxor is cooled from a high temperature, then the Burns temperature is a measure of when slow fluctuations become evident. These fluctuations are connected with the disorder and are known as nano-domains. The Burns temperature is not a well-defined transition temperature. At a lower temperature, there is a well-defined boundary to a so-called random field state when the nano-domains become static but there is no long-range periodic order. This phase may have both history-dependent properties and a skin effect in which the surface of the sample is different from that of the bulk material, as also found in experiments on magnetic systems. Section 1 is an introduction to the review, to ferroelectricity and to relaxors. Section 2 gives a description of the results obtained by dielectric, optical, specific heat and other macroscopic properties. These long-wavelength properties give a variety of different characteristic temperatures and do not directly probe the random field state. In Section 3, we describe the results of neutron and X-ray scattering and show that they strongly support the interpretation that relaxors have a random field state. In Section 4, we briefly describe the results for other relaxor systems such as (PMN)_1−x (PT) _x for which PMN is mixed with different amounts of the ferroelectric lead titanate (PT), and we show that the existence of a random field state enables us also to describe the experimental results for these mixed materials. We hope that this review will inspire further theoretical and experimental work to understand the nature of the random field states and to compare the experimental results more satisfactorily with theory.